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Programming for GCSE
Topic 2.3: Converting Binary and Decimal
Teaching London Computing
William MarshSchool of Electronic Engineering and Computer Science
Queen Mary University of London
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Aims
• Understand powers of 2• Number of numbers
• Convert between binary and decimal• Using powers of 2
• Do binary arithmetic with more interpretation• How fixed width leads to overflow in
addition
• Look at some pictures• Notes on the syllabus
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Teaching Issue
• So far, used counting … • … now more mathematics.
• Still, minimise notation and terms
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CONVERTING BETWEEN BINARY AND DECIMAL
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Decimal – base 10
• Base 10• 10 numerals• ‘0’, ‘1’, ‘2’, … , ‘9’
• What does ‘123’ mean?
123 = 1 x 100 + 2 x 10 + 3 x 1
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Base 10 Table
110100
1 2 3
mostsignificant digit
leastsignificant digit
123 = 1 x 100 + 2 x 10 + 3 x 1
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Base 2 Table
124816
1 0 1 0 0
mostsignificant bit
leastsignificant bit
10100 = 1 x 16 + 0 x 8 + 1 x 4 + 0 x 2 + 0 x 110100 = 16 + 4 = 20
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Conversion to Binary
• To convert a decimal integer to binary:• Odd 1, Even 0• Divide by 2• Stop when result of the division is 0
123 61 30 15 7 3 1 0
MostSignificantbit01 1 1 1 1 1
12310 = 1 1 1 1 0 1 12
LeastSignificant
bit
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Quiz
• Convert 00112 to decimal
• Convert 11112 to decimal
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POWERS AND EXPONENTS
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Powers and Exponents• 10N Power of 10
‘N’ is an exponent
• 100 = 1• 10(X+Y) = 10X x 10Y
100 = 1
101 = 10
102 = 10 x 10
103 = 10 x 10 x 10
21 = 1
21 = 2
22 = 2 x 2
23 = 2 x 2 x 2
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Base 2 Table
2021222324
1 0 1 0 0
mostsignificant bit
leastsignificant bit
10100 = 1 x 24 + 0 x 23 + 1 x 22 + 0 x 21 + 0 x 20
= 1 x 16 + 0 x 8 + 1 x 4 + 0 x 2 + 0 x 1
= 16 + 4 = 20
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Quiz
• Write out powers of 2, up to 28 (then 216)
• Convert 101000112 to decimal
• Convert 011111112 to decimal
• What is the next number after (the 10 digit number) 11111111112 in base 10?
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K and 210 and 103
• 210 = 1024, approximately equal to 103
• 210 abbreviated by ‘K’• 1 KByte is 1024 Bytes
• 220 = 210 x 210 103 x 103 = 106 • 220 abbreviated ‘M’ • 1 MByte = 220 Bytes 106 Bytes
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Quiz
• Which is larger• 232 or• the number of people in the world?
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HOW MANY NUMBERS
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How Many Numbers?
• How many decimal numbers less that 100?• 2-digit numbers : NN• 0 .. 99• 100 different numbers
• General rule: • 10n n-digits (decimal) numbers• 2n n-digit (binary) numbers
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How Many Binary Numbers?
bits max binary max base10 how many
1 1 1 22 11 3 43 111 7 84 1111 15 165 11111 31 326 111111 63 647 1111111 127 1288 11111111 255 256
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Quiz
• 161 student in this class How many bits to represent each student
with a unique binary number?
• A computer can execute 9 different machine instructions: ADD, SUB, MUL, DIV, JUMP, LOAD, READ, WRITE, STOP.
How many bits do we need to give each instruction a different code?What could these codes be?
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Quiz – Answers
• 7 bits?? • NO! With 7 bits we can only represent 27 = 128
patterns. • We need 8 bits. 8 bits can represent up to 28 =
256 patterns
• To represent 9 bit patterns we need 4 bits: 24 = 16
0000 ADD 0001 SUB 0010 MUL 0011 DIV 0100 JUMP
0101 LOAD 0110 READ0111 WRITE1111 STOP
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Overflow
Arithmetic with Fixed Number of Digits
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Overflow
• The addition of 8 bit numbers may overflow 8 bits
• Computer arithmetic has a limited number of bits
1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0
+
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Fixed Bit Arithmetic
• 4-bit0000
0001
0010
0011
0100
0101
0110
011110001001
1010
1011
1100
1101
1110
1111
01
2
3
4
5
6
78
910
11
12
13
1415
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Fixed Bit Arithmetic
• 4-bit
• Add 4
00000001
0010
0011
0100
0101
0110
011110001001
1010
1011
1100
1101
1110
1111
0 0 0 0 0 1 0 0 0 1 0 0
+
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Fixed Bit Arithmetic
• 4-bit
• Add 4
00000001
0010
0011
0100
0101
0110
011110001001
1010
1011
1100
1101
1110
1111
0 1 1 1 0 1 0 0 1 0 1 1
+
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Fixed Bit Arithmetic
• 4-bit
• Add 4
00000001
0010
0011
0100
0101
0110
011110001001
1010
1011
1100
1101
1110
1111
1 1 1 0 0 1 0 0 1 0 0 1 0
+
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Overflow (Unsigned)
• When you passed the read line
• E.g.• 14 + 4 = 2
00000001
0010
0011
0100
0101
0110
011110001001
1010
1011
1100
1101
1110
1111
01
2
3
4
5
6
78
910
11
12
13
1415
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IMAGES
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Two Ideas – Images
• Pixels and resolution• Image is an array of pixels
• Number of bits per pixel• ‘Colour’ of each pixel is a number
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Original
• Red – 8 bits• Green – 8 bits• Blue – 8 bits• ‘Million’
colours
508 × 578 pixels24 bit RGB colour
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Fewer Pixels
• m100 × 114 pixels24 bit RGB colour
50 × 57 pixels24 bit RGB colour
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Fewer Colours
508 × 578 pixels24 bit RGB colour
508 × 578 pixels256 colours (indexed)
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SYLLABUS
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Syllabus – Binary
• GCSE (OCR)• Conversion between binary and decimal• Hexadecimal• Binary addition
• AS/A2 (AQA)• (AS) Negative numbers - two’s complement • (AS) More arithmetic• (A2) Real (floating point) numbers
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Summary
• Understand powers of 2• How many bits binary representation• Arithmetic with fixed number of bits leads to
overflow
• Images• Pixels• Bits per pixel
• Anything can be represented by numbers (i.e. digitally)